Lineal-path function for random heterogeneous materials

A fundamental morphological measure of two-phase heterogenous materials is what we refer to as the lineal-path function L(z). This quantity gives the probability that a line segment of length z is wholly in one of the phases, say phase 1, when randomly thrown into the sample. For three-dimensional systems, we observe that L(z) is also equivalent to the area fraction of phase 1 measured from the projected image onto a plane: a problem of long-standing interest in stereology. We develop a theoretical means of representing and computing the lineal-path function L(z) for distributions of D-dimensional spheres with arbitrary degree of penetrability using statistical-mechanical concepts. In order to test our theoretical results, we determined L(z) from Monte Carlo simulations for the case of three-dimensional systems of spheres and found very good agreement between theory and the Monte Carlo calculations.